A226738 -id:A226738 - cp's OEIS frontend

A226513 Array read by antidiagonals: T(n,k) = number of barred preferential arrangements of k things with n bars (k >=0, n >= 0).

1, 1, 1, 1, 2, 3, 1, 3, 8, 13, 1, 4, 15, 44, 75, 1, 5, 24, 99, 308, 541, 1, 6, 35, 184, 807, 2612, 4683, 1, 7, 48, 305, 1704, 7803, 25988, 47293, 1, 8, 63, 468, 3155, 18424, 87135, 296564, 545835, 1, 9, 80, 679, 5340, 37625, 227304, 1102419, 3816548, 7087261

Offset: 0

N. J. A. Sloane, Jun 13 2013

The terms of this sequence are also called high-order Fubini numbers (see p. 255 in Komatsu). - Stefano Spezia, Dec 06 2020

			Array begins:
  1  1   3   13    75    541     4683     47293     545835 ...
  1  2   8   44   308   2612    25988    296564    3816548 ...
  1  3  15   99   807   7803    87135   1102419   15575127 ...
  1  4  24  184  1704  18424   227304   3147064   48278184 ...
  1  5  35  305  3155  37625   507035   7608305  125687555 ...
  1  6  48  468  5340  69516  1014348  16372908  289366860 ...
  ...
Triangle begins:
  1,
  1, 1,
  1, 2, 3,
  1, 3, 8, 13,
  1, 4, 15, 44, 75,
  1, 5, 24, 99, 308, 541,
  1, 6, 35, 184, 807, 2612, 4683,
  1, 7, 48, 305, 1704, 7803, 25988, 47293,
  1, 8, 63, 468, 3155, 18424, 87135, 296564, 545835
  ........
[_Vincenzo Librandi_, Jun 18 2013]

Z.-R. Li, Computational formulae for generalized mth order Bell numbers and generalized mth order ordered Bell numbers (in Chinese), J. Shandong Univ. Nat. Sci. 42 (2007), 59-63.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
Toka Diagana and Hamadoun Maïga, Some new identities and congruences for Fubini numbers, J. Number Theory 173 (2017), 547-569.
Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263 (p. 255).

Rows 0-5 give: A000670, A005649, A226515, A226738, A226739, A226740.
Columns 2, 3 = A005563, A226514.
Cf. A053492 (array diagonal), A265609, A346982.

T:= (n, k)-> k!*coeff(series(1/(2-exp(x))^(n+1), x, k+1), x, k):
seq(seq(T(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Mar 26 2016

T[n_, k_] := Sum[StirlingS2[k, i]*i!*Binomial[n+i, i], {i, 0, k}]; Table[ T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2016 *)

T(n,k) = Sum_{i=0..k} S2_k(i)*i!*binomial(n+i,i), where S2_k(i) is the Stirling number of the second kind. - Jean-François Alcover, Mar 26 2016
T(n,k) = k! * [x^k] 1/(2-exp(x))^(n+1). - Alois P. Heinz, Mar 26 2016
Conjectural g.f. for row n as a continued fraction of Stieltjes type: 1/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 4*x/(1 - (n+3)*x/(1 - 6*x/(1 - ... ))))))). Cf. A265609. - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
T(n,0) = 1; T(n,k) = Sum_{j=1..k} (n*j/k + 1) * binomial(k,j) * T(n,k-j).
T(n,0) = 1; T(n,k) = (n+1)*T(n,k-1) - 2*Sum_{j=1..k-1} (-1)^j * binomial(k-1,j) * T(n,k-j). (End)
G.f. for row n: (1/n!) * Sum_{m>=0} (n+m)! * x^m / Product_{j=1..m} (1 - j*x), for n >= 0. - Paul D. Hanna, Feb 01 2024

A226515 Row 2 of array in A226513.

Original entry on oeis.org

1, 3, 15, 99, 807, 7803, 87135, 1102419, 15575127, 242943723, 4145495055, 76797289539, 1534762643847, 32907617073243, 753473367606975, 18347287182129459, 473409784213526967, 12902366605394652363, 370357953441110390895, 11167936445234485414179

Offset: 0

N. J. A. Sloane, Jun 13 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 3, 4, 5 of A226513: A000670, A005649, A226738, A226739, A226740.

m:=2; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]]; // Bruno Berselli, Jun 18 2013

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-3, {x, 0, 20}], x] (* Vincenzo Librandi, Jun 18 2013 *)

E.g.f.: 1/(2 - exp(x))^3 (see the Ahlbach et al. paper, Theorem 4). - Vincenzo Librandi, Jun 18 2013
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(2+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 18 2013]
G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(k + 1) - 2*x^2*(k + 1)*(k + 3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2013
G.f.: 1/(1 + x)/Q(0,u), where u = x/(1 + x), Q(k,u) = 1 - u*(3*k + 4) - 2*u^2*(k + 1)*(k + 3)/Q(k+1,u); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n^2 /(16*(log(2))^(n + 3)) * (1 + 3*(1 + log(2))/n). - Vaclav Kotesovec, Oct 08 2013
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - (n+2)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

More terms from Vincenzo Librandi, Jun 18 2013

A226739 Row 4 of array in A226513.

Original entry on oeis.org

1, 5, 35, 305, 3155, 37625, 507035, 7608305, 125687555, 2265230825, 44210200235, 928594230305, 20880079975955, 500343586672025, 12726718227077435, 342425052939060305, 9715738272696568355, 289901469137229041225, 9074304882434034258635, 297297854264669632338305

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 3 and 5 of A226513: A000670, A005649, A226515, A226738, A226740.

m:=4; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];

Range[0,20]! CoefficientList[Series[(2 - Exp@x)^-5, {x, 0, 20}],x]

E.g.f.: 1/(2 - exp(x))^5 (see the Ahlbach et al. paper, Theorem 4).
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(4+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k + 1 + m) - 2*x^2*(k + 1)*(k + 1 + m)/Q(k+1), m = 4 is row 4 of array in A226513; (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n^4 / (768 * log(2)^(n+5)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - (n+4)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (4*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A226740 Row 5 of array in A226513.

Original entry on oeis.org

1, 6, 48, 468, 5340, 69516, 1014348, 16372908, 289366860, 5553635436, 114964523148, 2552305112748, 60474398655180, 1522843616043756, 40605864407444748, 1142786353739186988, 33848016050071188300, 1052381222812017946476, 34266937867683980363148, 1166071764343727862515628

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 3, 4 of A226513: A000670, A005649, A226515, A226738, A226739.

m:=5; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-6, {x, 0, 20}], x]

E.g.f.: 1/(2 - exp(x))^6 (see the Ahlbach et al. paper, Theorem 4).
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(5+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
a(n) ~ n! * n^5 / (7680 * log(2)^(n+6)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 8*x/(1 - 6*x/(1 - (n+5)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (5*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A232473 3-Fubini numbers.

Original entry on oeis.org

6, 42, 342, 3210, 34326, 413322, 5544342, 82077450, 1330064406, 23428165002, 445828910742, 9116951060490, 199412878763286, 4646087794988682, 114884369365147542, 3005053671533400330, 82905724863616146966, 2406054103612912660362, 73277364784409578094742, 2336825320400166931304970

Offset: 3

N. J. A. Sloane, Nov 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 3..200
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.

Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A226738, A232472, A232473, A232474.

r:=3; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

# r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(3);

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 3], {n, 3, 22}] (* Jean-François Alcover, Mar 30 2016 *)

From Peter Bala, Dec 16 2020: (Start)
a(n+3) = Sum_{k = 0..n} (k+3)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n ).
a(n+3) = Sum_{k = 0..n} 3^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+3)! ).
E.g.f. with offset 0: 6*exp(3*z)/(2 - exp(z))^4 = 6 + 42*z + 342*z^2/2! + 3210*z^3/3! + .... (End)
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020

A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

Original entry on oeis.org

1, 4, 16, 68, 316, 1616, 9080, 55800, 373080, 2699520, 21035040, 175708320, 1566916320, 14862171840, 149429426880, 1587766126080, 17779538050560, 209295747832320, 2583920845209600, 33389139008678400, 450642388471395840, 6342869733912760320

Offset: 0

Seiichi Manyama, May 17 2022

sign

a(46) is negative. - Vaclav Kotesovec, Jun 04 2022

It appears that a(n) is negative for even n >= 46. - Felix Fröhlich, Jun 04 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..450

Cf. A006252, A317280, A354120.

Cf. A226738, A354123.

Table[Sum[(k+3)! * StirlingS1[n,k], {k,0,n}]/6, {n,0,20}] (* Vaclav Kotesovec, Jun 04 2022 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^4))

a(n) = sum(k=0, n, (k+3)!*stirling(n, k, 1))/6;

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354123 Expansion of e.g.f. 1/(1 + log(1 - x))^4.

Original entry on oeis.org

1, 4, 24, 188, 1804, 20416, 265640, 3901320, 63776280, 1147796160, 22540858080, 479500074720, 10980929163360, 269298981833280, 7040446188020160, 195439047629422080, 5740498087530831360, 177855276360034736640, 5796391124741936993280

Offset: 0

Seiichi Manyama, May 17 2022

nonn

Cf. A007840, A052801, A354122.

Cf. A226738, A354121.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^4))

a(n) = sum(k=0, n, (k+3)!*abs(stirling(n, k, 1)))/6;

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * |Stirling1(n,k)|.
a(n) ~ sqrt(Pi/2) * n^(n + 7/2) / (3 * (exp(1) - 1)^(n+4)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (3*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A377399 Expansion of e.g.f. (2 - exp(x))^4.

Original entry on oeis.org

1, -4, 8, 8, -40, -184, -232, 1928, 19160, 116936, 600728, 2826248, 12623960, 54550856, 230564888, 959736968, 3952166360, 16149893576, 65626404248, 265592398088, 1071642518360, 4314414017096, 17341238230808, 69615800073608, 279215943071960, 1119122403273416

Offset: 0

Seiichi Manyama, Oct 27 2024

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A226738.

a(n) = sum(k=0, 4, (-1)^k*k!*binomial(4, k)*stirling(n, k, 2));

a(n) = sum(k=0, 4, (-1)^k*2^(4-k)*binomial(4, k)*k^n);

a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3) + 192 for n > 3.
a(n) = Sum_{k=0..4} (-1)^k * k! * binomial(4,k) * Stirling2(n,k).
a(n) = Sum_{k=0..4} (-1)^k * 2^(4-k) * binomial(4,k) * k^n.
G.f.: (1-14*x+83*x^2-262*x^3+384*x^4)/((1-x) * (1-2*x) * (1-3*x) * (1-4*x)).
a(n) = 4^n - 8*3^n + 3*2^(n+3) - 32 for n > 0. - Stefano Spezia, Oct 27 2024
a(0) = 1; a(n) = Sum_{k=1..n} (1 - 5 * k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Oct 27 2024

cp's OEIS Frontend

A226513 Array read by antidiagonals: T(n,k) = number of barred preferential arrangements of k things with n bars (k >=0, n >= 0).

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A226515 Row 2 of array in A226513.

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A226739 Row 4 of array in A226513.

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A226740 Row 5 of array in A226513.

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A232473 3-Fubini numbers.

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A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

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A354123 Expansion of e.g.f. 1/(1 + log(1 - x))^4.

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A377399 Expansion of e.g.f. (2 - exp(x))^4.

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